Problem: Find the greatest common factor of $50, 25,$ and $100$.
The greatest common factor (GCF) is the largest number that is a factor of $50, 25,$ and $100$. In order to find the GCF, we can factor each number completely as a product of prime numbers: $ \begin{aligned}50 &=2\cdot5\cdot5\\\\\\\\ 25&=5\cdot5\\\\\\\\ 100&=2\cdot2\cdot5\cdot5 \end{aligned}$ Now, let's find the factors that are common to each number: $ \begin{aligned}50 &=2\cdot5\cdot5\\\\\\\\ 25&=5\cdot5\\\\\\\\ 100&=2\cdot2\cdot5\cdot5 \end{aligned}$ Each number shares the factors ${5}$ and $5,$ so the GCF is $5\cdot5={25}$. The greatest common factor of $50, 25,$ and $100$ is $25$.